Reciprocal Graphs

Reciprocal Graphs are graphical representations of reciprocal Functions generically represented as $y = \frac{a}{x}$ and $y = \frac{a}{x^{2}}$ , where the numerator $a$ is a real constant, and the denominator contains an algebraic expression with a variable x.

Get started

Fact Checked Content
Last Updated: 01.12.2023
8 min reading time

Content creation process designed by
Content cross-checked by
Content quality checked by

Using reciprocal graphs

Reciprocal Graphs are useful to visually represent relationships that are inversely proportional, which means that they behave in opposite ways – if one decreases, the other one increases, and vice versa. For example, if the Number of workers in a shop increases, the amount of time that the customers spend waiting to be served will be reduced.

Asymptotes

To sketch this type of graph, you need to take into account its asymptotes. An asymptote is a line that the curve gets very close to, but never touches. The graph of reciprocal Functions $y = \frac{a}{x}$ and $y = \frac{a}{x^{2}}$ have asymptotes at $x = 0$ and $y = 0$ .

See the graph below for $y = \frac{1}{x}$

Reciprocal Graphs Reciprocal function asymptotes StudySmarter Reciprocal function asymptotes, Marilú García De Taylor - StudySmarter Originals

$x = 0$ is a vertical asymptote because you cannot divide by zero; therefore, x cannot be zero. $y = 0$ is a horizontal asymptote because there are no values of x that make $y = 0$ , so y cannot be zero either.

y=0Notice that the graph of $y = \frac{1}{x}$ is symmetric to the lines $y = x$ and $y = - x$ .

y=0

y=x Reciprocal Graphs Reciprocal function symmetry StudySmarter Reciprocal function y = 1 / x - symmetry to y = x, Marilú García De Taylor - StudySmarter Originals

Reciprocal Graphs Reciprocal function symmetry StudySmarter Reciprocal function y = 1 / x - symmetry to y = -x, Marilú García De Taylor - StudySmarter Originals

In general, the domain of reciprocal functions will be all Real Numbers apart from the vertical asymptote, and the range will be all Real Numbers apart from the horizontal asymptote.

Types of reciprocal graphs

The Graphs article discusses that the coordinate plane is divided into four quadrants named using roman numbers (I, II, III and IV):

Reciprocal Graphs Coordinate plane StudySmarter Coordinate plane, Marilú García De Taylor - StudySmarter Originals

The possible types of reciprocal graphs include:

Reciprocal functions of the type $y = \frac{a}{x}$

a) If a> 0:

For example, if $a = 1$ , $y = \frac{1}{x}$ , the shape of the graph is shown below. Notice that the graph is drawn on quadrants I and III of the coordinate plane.

Reciprocal Graphs Reciprocal function positive numerator StudySmarter Reciprocal function, Marilú García De Taylor - StudySmarter Originals

b) If a <0:

For example, if $a = - 1$ , $y = \frac{- 1}{x}$ , the shape of the reciprocal function is shown below. In this case, the graph is drawn on quadrants II and IV. This graph is the reflection of the previous one because the negative sign in the function means that all positive values of $x \neq 0$ will now have negative values of y, and all negative values of x will now have positive values of y.

Reciprocal Graphs Reciprocal function negative numerator StudySmarter Reciprocal function with negative numerator, Marilú García De Taylor - StudySmarter Originals

Reciprocal functions of the type $y = \frac{a}{x^{2}}$

a) If a> 0:

For example, if $a = 1$ , $y = \frac{1}{x^{2}}$ , the shape of the graph is shown below. Notice that the graph is drawn on quadrants I and II of the coordinate plane. The shape of the graph of $y = \frac{1}{x^{2}}$ changes in comparison to the previous graph of $y = \frac{1}{x}$ , because having $x^{2}$ in the denominator means that all values of y will be positive for all values of $x \neq 0$ .

Reciprocal Graphs Reciprocal squared function StudySmarter Reciprocal squared function, Marilú García De Taylor - StudySmarter Originals

b) If a <0:

For example, if $a = - 1$ , $y = \frac{- 1}{x^{2}}$ , the shape of the reciprocal function is shown below. In this case, the graph is drawn on quadrants III and IV. This graph is also the reflection of the previous one due to the negative sign in the numerator of the function.

Reciprocal Graphs Reciprocal squared function negative numerator StudySmarter Reciprocal squared function with negative numerator, Marilú García De Taylor - StudySmarter Originals

Drawing reciprocal graphs

To show you how to draw the graph of a reciprocal function, we will use the example of $y = \frac{1}{x}$ . To graph this function you need to follow these steps:

Identify the vertical and horizontal asymptotes.

For $y = \frac{1}{x}$ , $x = 0$ and $y = 0$ are asymptotes.

Identify the type of reciprocal function $y = \frac{a}{x}$ or $y = \frac{a}{x^{2}}$ , and if a is positive or negative. This information will give you an idea of where the graphs will be drawn on the coordinate plane. This step is optional.

In our example $y = \frac{1}{x}$ , the reciprocal function is of type y = $\frac{a}{x}$ and a> 0; therefore, the graphs will be drawn on quadrants I and III.

Plot points strategically reveal the graph's behaviour as it approaches the asymptotes from each side.

Negative side:

$f (- 1) = \frac{1}{- 1} = - 1$

$f (- 2) = \frac{1}{- 2} = \frac{- 1}{2}$

$f (- 3) = \frac{1}{- 3} = \frac{- 1}{3}$

Notice that the further we go to the left, the closer we get to zero.

Now let's try some fractions of negative 1:

$f (\frac{- 1}{2}) = \frac{1}{\frac{- 1}{2}} = - 2$

$f (\frac{- 1}{3}) = \frac{1}{\frac{- 1}{3}} = - 3$

Positive side:

$f (1) = \frac{1}{1} = 1$

$f (2) = \frac{1}{2}$

$f (3) = \frac{1}{3}$

Notice that the further we go to the right, the closer we get to zero.

Now let's try some fractions of positive 1:

$f (\frac{1}{2}) = \frac{1}{\frac{1}{2}} = 2$

$f (\frac{1}{3}) = \frac{1}{\frac{1}{3}} = 3$

x	$- 3$	$- 2$	$- 1$	$\frac{- 1}{2}$	$\frac{- 1}{3}$	$\frac{1}{3}$	$\frac{1}{2}$	$1$	$2$	$3$
y	$\frac{- 1}{3}$	$\frac{- 1}{2}$	$- 1$	$- 2$	$- 3$	$3$	$2$	$1$	$\frac{1}{2}$	$\frac{1}{3}$

Draw the graph using the table of values obtained:

Reciprocal Graphs Reciprocal function graph StudySmarter Reciprocal function graph, Marilú García De Taylor - StudySmarter Originals

A reciprocal function $y = \frac{a}{x}$ has been transformed if its equation is written in the Standard Form $y = \frac{a}{x + h} + k$ , where a, h and k are real constants, the vertical asymptote of the function is $x = - h$ , and the horizontal one is $y = k$ .

For the reciprocal function $y = \frac{1}{x + 2} + 1$ , the asymptotes are $x = - 2$ and $y = 1$ .

Reciprocal Graphs Reciprocal graph equation standard form StudySmarter Reciprocal graph with the equation in standard form, Marilú García De Taylor - StudySmarter Originals

You might be asked to find the interceptions of the reciprocal function graph with the x and y axes. You can proceed as follows:

x-intercept: Substitute y = 0 in the equation and solve for x.

$0 = \frac{1}{x + 2} + 1$

$- 1 = \frac{1}{x + 2}$

$- (x + 2) = 1$

$- x - 2 = 1$

$x = - 2 - 1$

$x = - 3$

The point where the graph of the function crosses the x-axis is (-3, 0)

y-intercept: Substitute x = 0 in the equation and solve for y.

$y = \frac{1}{0 + 2} + 1$

$y = \frac{1}{2} + 1$

$y = \frac{3}{2}$

The point where the graph of the function crosses the y-axis is $(0, \frac{3}{2})$

How do you find the equation of a reciprocal graph?

If you are given a reciprocal graph, you can find its equation $y = \frac{a}{x + h} + k$ by following these steps:

Find the vertical asymptote. This is the value you need to add or subtract from the variable in the denominator $(h)$ . It will have the opposite sign of the vertical asymptote.
Find the horizontal asymptote. This will be the value of $k$ , which is added or subtracted from the fraction depending on its sign.
Find the value of $a$ by substituting the x and y corresponding to a given point on the curve in the equation.

Find the equation for the reciprocal graph below:

Reciprocal Graphs Equation of a reciprocal graph StudySmarter Equation of a reciprocal graph, Marilú García De Taylor - StudySmarter Originals

Vertical asymptote $x = - 3$ , therefore $h = 3$
Horizontal asymptote $y = 1$ , therefore $k = 1$
Substitute the point A (-5, 0) in the reciprocal function $y = \frac{a}{x + h} + k$ to find the value of $a$ :

$0 = \frac{a}{- 5 + 3} + 1$

$0 = \frac{a}{- 2} + 1$

$- 1 = \frac{a}{- 2}$

$a = 2$

The equation of the reciprocal function is $y = \frac{2}{x + 3} + 1$

Finding the reciprocal of a function

We know from Algebra that you can calculate the reciprocal of a Number by swapping the numerator and the denominator. The same applies to functions. To find the reciprocal of a function $f (x)$ you can find the expression $\frac{1}{f (x)}$ .

Find the reciprocal of the function $y = x - 5$

The reciprocal of $y = x - 5$ is $y = \frac{1}{x - 5}$

Reciprocal Graphs - Key takeaways

Reciprocal graphs are graphical representations of reciprocal functions, where the numerator is a real constant, and the denominator contains an algebraic expression with a variable x.
To sketch reciprocal graphs, you must take into account their asymptotes.
An asymptote is a line that the curve gets very close to, never touching it.
The domain of reciprocal functions will be all real numbers apart from the vertical asymptote.
The range of reciprocal functions will be all real numbers apart from the horizontal asymptote.

Already have an account? Log in

Frequently Asked Questions about Reciprocal Graphs

What is a reciprocal graph?

Reciprocal graphs are graphical representations of reciprocal functions, where the numerator is a real constant, and the denominator contains an algebraic expression with a variable x.

How do you draw a reciprocal graph?

To graph this function you need to follow these steps:

Identify the vertical and horizontal asymptotes.
Identify the type of reciprocal function y = a/x or y = a/x², and if a is positive or negative. This information will give you an idea of where the graphs will be drawn on the coordinate plane. (Optional)
Plot points strategically to reveal the behaviour of the graph as it approaches the asymptotes from each side.
Draw the graph using the table of values obtained.

How do you find the equation of a reciprocal graph?

To find the equation of a reciprocal function y = a/(x+h) + k follow these steps:

Find the vertical asymptote. This is the value that you need to add or subtract from the variable in the denominator (h). h will have the opposite sign of the vertical asymptote.
Find the horizontal asymptote. This will be the value of k, which is added or subtracted from the fraction depending on its sign.
Find the value of a by substituting the values of x and y corresponding to a given point on the curve in the equation.

How do you find the reciprocal of a function?

To find the reciprocal of a function f(x) you can find the expression 1/f(x).

Why do we use reciprocal graphs?

Reciprocal graphs are useful to visually represent relationships that are inversely proportional, which means that they behave in opposite ways. If one decreases the other one increases, and vice versa.

Save Article

Discover learning materials with the free StudySmarter app

About StudySmarter

StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

Learn more